发表机构
National Cheng Kung University; National Yang Ming Chiao Tung University; University of Duisburg-Essen（国立成功大学; 国立阳明交通大学; 杜伊斯堡-埃森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究三维及以上光滑有界区域上加权p-Laplace方程的刚性，证明接近1的权重若在某仿射边界值的Dirichlet-to-Neumann映射导数与单位权重一致则权重恒为1，并证明任意方向恒定的参考权重处的局部刚性，结合线性化与各向异性Calderón问题结果。
AI 中文摘要
我们研究了三维及以上维数光滑有界区域上加权p-Laplace方程的刚性。我们证明，若一个光滑正权重充分接近1，且其在一个仿射边界值处的Dirichlet-to-Neumann映射的导数与单位权重的一致，则该权重恒等于1。我们还证明了在任意一个方向恒定的光滑正参考权重处的局部刚性。参考权重不必接近1，未知权重仅需充分接近参考权重，且无方向限制。不假设凸性、解析性或边界附近的吻合性。证明将无临界点解的线性化与各向异性Calderón问题的最新刚性结果相结合。线性化电导率张量的恒等式消除了剩余的微分同胚。最后一步不要求任一权重的小性。
英文摘要
We study rigidity for the weighted $p$-Laplace equation on smooth bounded domains in dimensions three and higher. We show that a smooth positive weight sufficiently close to $1$ is identically $1$ if the derivative of its Dirichlet-to-Neumann map at one affine boundary value agrees with that for the unit weight. We also prove local rigidity at every smooth positive reference weight which is constant in one direction. The reference weight need not be close to $1$, and the unknown weight is only required to be sufficiently close to it, without a directional restriction. No convexity, analyticity, or agreement near the boundary is assumed. The proof combines linearization at a solution without critical points with recent rigidity results for the anisotropic Calderón problem. An identity for the linearized conductivity tensors removes the remaining diffeomorphism. This last step does not require smallness of either weight.
